On solvability of the jump problem

In this paper we introduce an alternative way of defining the curvilinear Cauchy integral over non-rectifiable curves in the case of complex functions of one complex variable. Especially the jump behavior on the boundary is considered. As an application, solvability conditions of the Riemann boundary value problem are derived on very weak conditions to the boundary. Besides the complex case the consideration can be extended to the theory of Douglis algebra valued functions.

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“…The proof of inequalities (12) repeats that of the first proposition of Theorem 2 in [17]. Let us consider the rectangles R n = z = x + iy:…”

Section: Approximate Dimensionmentioning

confidence: 84%

“…The solvability of jump problem under assumption g ∈ H + (C, 1 2 Dma Γ ) is proved in [17]. Here we prove the representability of the solution as the Cauchy integral.…”

Section: Definitionmentioning

confidence: 86%

Integration over non-rectifiable curves and Riemann boundary value problems

Blaya

Reyes

Kats

2011

Journal of Mathematical Analysis and Applications

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“…Inequalities (6) are proved in precisely the same way as the inequality DmaΓ ≤ DmbΓ in [14]. The second assertion of the theorem can also be proved by the same argument as in [14], but we apply a somewhat different construction.…”

Section: Theorem 1 (I) For Any Plane Curve γmentioning

confidence: 90%

“…Let us show that they can be extend by continuity to larger spaces. For this purpose, we use the notion of the approximation dimension of a nonrectifiable curve introduced in [14].…”

mentioning

confidence: 99%

The Riemann boundary value problem on a closed nonrectifiable curve and the Cauchy transform

Kats

2011

Lobachevskii J Math

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The solutions of the Riemann boundary value problem on a closed nonrectifiable curve are represented as the Cauchy transforms of certain distributions.1. This paper is devoted to the following well-known boundary value problem. Let Γ be a closed Jordan curve in the complex plane C dividing the plane into a finite domain D + and a domain D − containing the infinite point. It is required to find a function Φ(z) holomorphic in C \ Γ, having boundary valuesvanishing at the infinite point, and satisfying the boundary conjugation conditionswhere G and g are given functions. This problem, known as the Riemann problem, has extensive applications. The classical theory of the Riemann problem (see [1,2]) is based on using the Cauchytype integralIn particular, for a piecewise smooth curve Γ, this integral, whose density f satisfies the H¨older condition with exponent ν ∈ (0, 1], gives a unique solution to the simplest case of the Riemann problem, namely, the jump problemIf the curve Γ is not rectifiable, then the integral over this curve is generally undefined, but the Riemann boundary value problem still makes sense. In the 1980s, we showed (see, e.g., [3]) that this problem is solvable if the boundary data G(t) and g(t) satisfy the H¨older condition with exponentwhere DmbΓ is the upper metric dimension (also known as Minkowski dimension, or box dimension; see [4,5]) of the curve Γ, which is defined byHere, N (ε, Γ) is the minimum number of disks of diameter ε covering the set Γ. However, representations of solutions in the form of contour integrals have not been obtained. In this paper, we obtain such representations.*

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“…The notions introduced below generalize to the higher dimensional framework the one of [49] (see also [15,21]). The perimeter of a finite polygonal domain P R nC1 , denoted by p.P /, is defined to be H n .…”

Section: An Approximate Settingmentioning

confidence: 99%